At this point I have built this circuit according to the schematic. It works well enough to drive single neon bulbs off a single AA cell, though I have only tested with 1.23V minimum as that is the lowest voltage that my crude bench supply can provide. For phosphor coated bulbs or with multiple bulbs connected it is usually necessary to increase the input to 1.5V or so.

When I increase the input voltage I get an increased HV output, which is nice. However, once I increase the input voltage high enough, the circuit will stop operating correctly and won't provide enough output voltage anymore. My goal right now is to understand why this happens and if possible, find a solution for this problem. It is nice that this circuit could run off a single cell battery, but I would like to be able to run it on better power sources if I want more output. I will try to get more information about this issue with my oscilloscope, and I will post an update with some scope captures.